Łukasiewicz logic and the divisible extension of probability theory

Abstract

We show that measurable fuzzy sets carrying the multivalued
Łukasiewicz logic lead to a natural generalization of the classical
Kolmogorovian probability theory. The transition from Boolean logic
to Łukasiewicz logic has a categorical background and the resulting
divisible probability theory possesses both fuzzy and quantum qualities.
In the divisible probability theory, morphisms are called observables
and play analogous role as classical random variables - convey stochastic
information from one system of random events to another one.
Observables preserving the Łukasiewicz logic are called conservative.
They are exactly observables that characterize the ``classical core''
of divisible probability theory. They send crisp random events to crisp
random events and Dirac probability measures to Dirac probability measures.
The nonconservative observables send some crisp random events to genuine fuzzy
events and some Dirac probability measures to nondegenerated probability
measures. They constitute the added value of transition from
classical to divisible probability theory.